Part 1. Determination of the Path of a Point of a Body

 

Let us now investigate the motion of individual points of a rigid body, i.e., determine their paths, velocities and accelerations. For this, as has been shown, it is sufficient to analyze the motion of the points lying in section S. We shall begin with the determination of the paths.

Consider a point M of a body whose position in section S is specified by its distance b = AM from the pole A and the angle BAM = α (Fig. 18). If the motion of the body is described by Eqs. (48), the x and y coordinates of point M in the system Oxy will be

                                                          (49)

where xA, yA,φ are the functions of time t given by Eqs. (48).

Fig. 18

 

Eqs. (49) describes the motion of point M in plane Oxy and at the same time gives the equation of the point’s path in parametric form. The usual equation of the path can be obtained by eliminating time t from Eqs. (49).

If the body under consideration is part of a mechanism, the path of any point M of the body can be determined by expressing the coordinates of the point in terms of a parameter specifying the position of the mechanism and then eliminating that parameter. In this case the equations of motion (48) are not necessary.

 

Comprehension  check.

Ex.1. Put the questions to the following answers.

1. To analyze the motion of the points lying in section S.

2. By the distance of a point M of a body from the pole A and the angle α.

3. It describes the motion of point M in plane Oxy.

Part 2. Determination of the Velocity of a Point of a Body

 

Plane motion of a rigid body is a combination of a translation in which all points of the body move with the velocity of the pole  and a rotation about that pole. Let us show that the velocity of any point M of the body is the geometrical sum of its velocities for each component of the motion.

The position of a point M in section S of the body is specified with reference to the coordinate axes Oxy by the radius vector   ( Fig.19), where  is the radius vector of the pole A,  is the vector which specifies the position of point M with reference to the axes Ax'y' that perform translational motion together with A (the motion of section S with reference to those axes is the motion about pole A). Then, .

Fig. 19                                Fig. 20

  In this equation  is equal to the velocity of pole A; the quantity  is equal to the velocity  of point M at ., i.e., when A is fixed or, in other words, when the body (or, strictly speaking, its section S) rotates about pole A. It thus follows from the preceding equation that

                                 .                       (50)

The velocity of rotation  of point M about pole A is

                                 ,                     (51)

where ω is the angular velocity of the rotation of the body. Thus, the velocity of any point M of a body is the geometrical sum of the velocity of any other point A taken as the pole and the velocity of rotation of point M about the pole.

The magnitude and direction of the velocity  are found by constructing a parallelogram (Fig. 20).